docs: state syncs, reactions don't

AGENTS.md gets the "State syncs; reactions don't" section under Syncing
boards — the actor-only rule, the file-carried backstops behind it, and
the two consequences you can see (a half-done side effect is nobody's to
finish automatically; ownership gates work launches). The Pull requests
section says which board opens one and how ↑ open PR asks afterwards, the
card-actions list gains ▸ take over and ↑ open PR, "Claiming a card" stops
saying the claim refuses nothing, and the merge & clean up paragraph
documents the origin-side merge — including the constraint the local path
hid: whoever clicks needs merge rights on the repo, not just push rights.

.env.example: BOARD_COMMIT_MOVES documents the claim as a launch gate,
BOARD_SYNC the actor-only reactions and the origin-side merge.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>

manager/core/.env.example

@@ -74,6 +74,12 @@ BOARD_FETCH_TIMEOUT=10
 # assignee added to a file by hand is still read and shown either way;
 # the gate only governs whether a move writes it. Anything but
 # empty/0/false/no/off turns it on.
+#
+# It is also what makes the claim mean something: on, ▸ start work refuses
+# a card someone else holds (▸ take over is the deliberate way in) and
+# claims an unheld one, and board-made edits to a card in place — the
+# **PR:** line — commit themselves too. Off, nothing writes an assignee,
+# so nothing reads one as a lock.
 BOARD_COMMIT_MOVES=
 
 # Sync boards through origin/main: the second half of team mode, and off
@@ -92,6 +98,14 @@ BOARD_COMMIT_MOVES=
 # worktrees and PRs, so local main advances only through the board and
 # origin.
 #
+# State syncs; reactions don't. A move that arrives over origin renders
+# and narrates on every board but triggers nothing — opening a PR belongs
+# to the board whose user made the move, and a review card left without
+# one carries an explicit ↑ open PR action instead of N boards guessing.
+# The same discipline moves merges to origin: merge & clean up runs
+# `gh pr merge` and lets the beat deliver the result, so local main only
+# ever fast-forwards. Whoever clicks needs merge rights on the repo.
+#
 # One board fetches twice a minute at the default interval; raise it on a
 # rate-limited or metered remote. Sync rides `origin` and `main` only.
 BOARD_SYNC=